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if $f(x)$ is in $F[x]$. $F$ is field of integer mod $p$. $p$ is prime and $f(x)$ is irreducible over $F$ of degree $n$ . prove that $F[x]/(f(x))$ is a field with $p^n$ elements.

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I have approached by considering F[x] as Euclidean Ring. Also I have considered the Field formed by Quotient group of – White Dwarf Nov 17 '12 at 6:52
up vote 1 down vote accepted

By the irreducibility of $f(x)$, the quotient $F[x]/(f(x))$ is a field.

Show that distinct polynomials in $F[x]$ of degree $<n$ are inequivalent modulo $(f(x))$. This will follow from the fact that the polynomial ring $F[x]$ is an Euclidean domain.

This reduces the problem to counting the number of polynomials of degree $<n$ over $F$, which is straightforward.

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